Most of the browsers that I've used to FTP would do the same as your Windows explorer. They do not function as file managers on the remote ftp site.
I second the suggestion of getting an FTP client.
Even then, it might not act like a file manager, at least in my experience.
For example, to move something from Folder A to Folder B, you may not be able to just drag and drop it from one to the other on the remote site; you might have to download it from Folder A, upload it to Folder B, then delete it from Folder A.